Physically Possible Wavefunctions
Intuitive Quantum Physics
p.129
Name:___________________________
By this point in the semester, we’ve not only named a few problems to study further (what do you mean,
objects have wave-like properties in some situations and particle-like properties in others?!), we’ve also
begun to build a toolbox to use when we try to address these problems. Today, we apply our ideas about
energy, probability, and curviness to the wave-like behavior of electrons. Remember, we’re building a
model using the simplest possible tools. We’re going to weave our ideas together and begin to develop a
model for the atom.
We have used the Schrödinger equation to find the shape of the wave function:
"k # (TE object " PE system ) # $( x ) = Curv $(x)
where "(x) is the wave function, TE is the total energy of the object, PE is the potential energy of the
system in which the object is located, and k is some positive constant.
!
I. Energy and Wavelength
A. What happens to the curviness of "(x) when the value of "(x) increases and everything else stays the
same? Is this consistent with what you found in last week’s lab?
B. Imagine the total energy of the object is doubled, while the potential energy of the system in which
the object is located is zero. Use the Schrödinger equation to determine what happens to Curv "(x) at
a given value of "(x).
! 2003-6, University of Maine Physics Education Research Laboratory. (200701)
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Physically Possible Wavefunctions
II. Board Meeting 1
To be sure that we’re all on the same page, we’ll have our first board meeting early. Answer the
following question:
The graph on the left below shows an s-function that describes this object in some region in space.
On the right-hand graph, sketch a picture graph that represents the wave function of the object when
the total energy of the object increases. (Hint: You may find it helpful to draw a large version of the
left-hand function below on your whiteboard, and use your ideas about driving down a curve from
last week’s tutorial to help you reason about the answer to this question.)
"(x)
"(x)
0
1
2
3
4
x
5
6
0
1
2
3
4
5
x
Examining your sketch of the s-function for increased total energy, what can you say about the
relationship between the energy of an object and the wavelength of its wave function? Does the
wavelength increase, decrease, or stay the same?
III. Another wave function
Imagine another object in a different system (which is still at a potential energy of zero) that is described
by an e-function shown on the graph on the left below. Sketch a picture graph that represents the wave
function of the object when the total energy of the object is decreased, and the potential energy of the
system remains the same.
"(x
)
"(x
)
0
1
2
3
4
5
x
6
0
1
2
3
4
5
x
6
6
Physically Possible Wavefunctions
p.131
IV. Using the Wave Function
A. Use the Schrödinger equation to help you fill in the following table. We filled in the first line…
(
)
Schrödinger Equation: "k # TE object " PE system # $( x ) = Curv $(x)
sign of
–k
sign of
(TEobject–PEsystem)!
sign of
"
sign of
Curv"
Does graph curve toward
or away from axis?
is it an
s- or e-function?
–
toward
s
–
+
+
–
+
–
–
–
+
–
–
–
1. Sketch a wave function picture graph for the case when TE > PE. Label the axes of your graph.
2. Sketch a wave function picture graph for the case when TE < PE. Label the axes of your graph.
3. There are four correct answers to question 2. Working with your group, be sure that you have
drawn all of them before moving on.
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Physically Possible Wavefunctions
B. Five scenarios are listed below. For the four picture graphs of the wave function of an object shown in
the table below, choose the scenario that best matches the wave function, and explain why you chose
each matching scenario. Each graph corresponds to a unique scenario. Again, the Schrödinger
equation is: "k # (TE object " PE system ) # $( x ) = Curv $(x)
a. The total energy of the object is 10 J, and the potential energy of the system is 5 J.
!
b. The total energy of the object is 10 J, and the potential energy of the system is 15 J.
c. The total energy of the object is 20 J, and the potential energy of the system is 10 J.
d. The total energy of the object is 15 J, and the potential energy of the system is 15 J.
e. The total energy of the object is 5 J, and the potential energy of the system is 15 J.
"(x)
"(x)
0
1
2
3
4
5
x
6
2.
1.
0
Scenario:
Scenario:
Reasoning:
Reasoning:
"(x)
"(x)
0
3.
4.
0
1
2
3
4
5
x
6
Scenario:
Scenario:
Reasoning:
Reasoning:
1
2
1
2
3
3
4
4
5
5
x
x
6
6
Physically Possible Wavefunctions
p.133
V. Probability Density and the Wave Function
In the probability tutorial, we defined probability density as
P(x) = Probability Density of being in a region = Probability of being in a region/length of region
A. Rearrange the equation to find an expression for the probability in terms of the probability density
and length.
B. Think back to your homework, with Sally running along. We have recreated the sketch of her
running. From J to K she is at one speed, and from K to L she has halved her speed. We have also
shown the graph of her probability density in the space below the path on which she runs.
1. In which region are you most likely
to find Sally? Explain.
J
10 m
K
10 m
L
P(x)
2/30
2. Use the equation you found in A to
find the probability of Sally being
between:
1/30
10 m
20 m
a. J and K
x
b. K and L
C. A student says, “The probability of being in a region is the same as the area between the probability
density line and the x-axis for that region.” In your groups, test whether or not the student is right for
the case of Sally running. Do you think the student is right in general?
D. What is the probability of finding Sally somewhere between J and L? On the probability density
graph above, color in the area that corresponds to this probability.
E. With your group, develop a method to find the total probability for any probability density graph.
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Physically Possible Wavefunctions
Imagine that we are able to draw a wave function, "(x). In such a situation, we can find the probability
density by finding |"(x)|2. In math terms, we write that P(x) = |"(x)|2. One of the rules about probability
is that the total probability of an event happening (no matter which way it happens) is 1 (or 100%). This
basic fact sets some pretty strong constraints on what sort of wave function can describe a physical
system, or a situation that could actually happen. If the total probability is infinity, the P(x) graph cannot
describe a physical system.
F. Examine the graphs shown below of the wave function (an s-function) and probability density for a
specific energy state of an object in an infinite square well. (We’ll work out later how we arrived at
this wave function; for now, we’re making general statements about probability density).
1.2
1.2
P(x) = |"(x)|2
"(x)
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
0.5
1
1.5
2
x
0
2.5
0
0.5
1
1.5
2
x
2.5
-0.2
1. If we take a measurement of the system, where is the object most likely to be found? (Are you
circling a location on the function or the axis? Be explicit about why you choose one over the
other...)
2. If you found the area under the P(x) curve, what should you get? (We won’t ask you to do it, but
we expect you to know the answer!) Explain in your group what this means in a physical sense,
not just a mathematical sense! Feel free to check your answers with the TA
Physically Possible Wavefunctions
p.135
3. The wave function for a different object in a different energy state is shown below.
1.2
1.5
P(x) = |"(x)|2
"(x)
1
1
0.8
0.5
0.6
0
0
0.5
1
1.5
2
x
2.5
0.4
0.2
-0.5
0
-1
0
0.5
1
1.5
2
x
2.5
-0.2
-1.5
4. Sketch what the probability density graph looks like for 0 < x < 2. You can generalize from the
graphs in part A or you can calculate what |"(x)|2 should be by taking numbers from the "(x)
graph. (Hint: What happens when you square a negative number?)
5. At what location(s) is the object most likely to be found? Circle the place(s). (Are they on the xaxis or on the curve? How do you know?)
6. Could this wave function describe a physical system? (Hint: What happens if you add up all the
probability densities?) Be sure to explain your reasoning, not just say yes or no.
7. Based on the results from Board Meeting 1, does the wave function from A or B have higher
energy?
G. Imagine another object whose wave function is an e-function.
25
600
P(x) = |"(x)|2
"(x)
500
20
400
15
300
10
200
5
100
0
0
0.5
1
1.5
2
x
0
2.5
0
0.5
1
1.5
2
x
2.5
1. Sketch what the probability density graph looks like for 0 < x < 2. Note the different vertical axes
on the graphs.
2. At what location(s) is the object most likely to be found?
3. Imagine that this wave function continues to behave in the way suggested by its graph. That is,
for negative x values, the wave function continues to get closer and closer to zero and for
positive values of x > 2, the wave function continues to get larger and larger. Could this wave
function describe a physical system? Explain your reasoning.
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Physically Possible Wavefunctions
H. Decide whether or not each of the picture graphs of wave functions below could describe an actual
physical system. Assume each wave function continues in either direction in the manner indicated by
its behavior at the edges, and think about the behavior of the wave function as x becomes a large
positive value and a large negative value.
•
•
If the wave function can describe an actual physical system, explain why.
If the wave function cannot describe an actual physical system, explain why not, giving as many
reasons as you can think of.
"(x)
"(x)
"(x)
x
x
1.
2.
x
3.
Physically Possible? Yes No
Physically Possible? Yes No
Physically Possible? Yes No
Reason(s):
Reason(s):
Reason(s):
"(x)
"(x)
"(x)
x
4.
x
5.
6.
Physically Possible? Yes No
Physically Possible? Yes No
Physically Possible? Yes No
Reason(s):
Reason(s):
Reason(s):
x